How can I solve this differential equation, what type is it?

Question

How can I solve this differential equation, what type is it? $$(x^2+2x-2y)dx=dy$$

How can I find the integrating factor?

Answer 1

HINT

If the equation is $$ (x^2+2x-2y)*dx=dy $$then you can rewrite it as $$ \dfrac{dy}{dx} + 2y = x^2 + 2x $$ Now, you find the integrating factor: $ \text{I.F. } = e^{\int 2 dx} = e^{2x} $

The result is given by: $$ y \cdot e^{2x} = \int{ (x^2 + 2x)e^{2x} dx} $$

Such equations are pretty common and are called first-order linear differential equations (Thanks to @bof for comment).